Nuprl Lemma : rmul-limit
∀x,y:ℕ ⟶ ℝ. ∀a,b:ℝ. (lim n→∞.x[n] = a ⇒ lim n→∞.y[n] = b ⇒ lim n→∞.x[n] * y[n] = a * b)
Proof
Definitions occuring in Statement :
converges-to: lim n→∞.x[n] = y,
rmul: a * b,
real: ℝ,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
converges-to: lim n→∞.x[n] = y,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
converges: x[n]↓ as n→∞,
bounded-sequence: bounded-sequence(n.x[n]),
nat_plus: ℕ+,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
nat: ℕ,
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
true: True,
sq-all-large: ∀large(n).{P[n]},
rneq: x ≠ y,
sq_exists: ∃x:{A| B[x]},
rsub: x - y,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
rleq: x ≤ y,
rnonneg: rnonneg(x)
Lemmas referenced :
converges-to_wf,
nat_wf,
real_wf,
integer-bound,
converges-implies-bounded,
imax_wf,
imax_nat_plus,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
equal_wf,
less_than_wf,
rleq_wf,
rabs_wf,
int-to-real_wf,
all_wf,
rleq-int,
imax_ub,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rabs-bounds,
nat_properties,
mul_bounds_1a,
false_wf,
multiply_nat_wf,
nat_plus_subtype_nat,
mul_nat_plus,
sq-all-large-and,
rsub_wf,
rdiv_wf,
rless-int,
mul_bounds_1b,
rless_wf,
rmul_wf,
radd_wf,
r-triangle-inequality,
rminus_wf,
uiff_transitivity,
rleq_functionality,
req_weakening,
rabs_functionality,
radd_functionality,
req_transitivity,
rmul-distrib,
rmul_over_rminus,
req_inversion,
radd-assoc,
radd-ac,
radd-rminus-assoc,
rabs-rmul-rleq,
itermMultiply_wf,
int_term_value_mul_lemma,
rleq-int-fractions,
sq_stable__all,
sq_stable__rleq,
less_than'_wf,
squash_wf,
rmul-int-rdiv2,
rmul-int-rdiv,
itermAdd_wf,
int_term_value_add_lemma,
uimplies_transitivity,
rdiv_functionality,
radd-int,
radd-rdiv,
radd_functionality_wrt_rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
hypothesis,
functionEquality,
dependent_functionElimination,
productElimination,
dependent_pairFormation,
independent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
independent_isectElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
productEquality,
because_Cache,
inrFormation,
inlFormation,
multiplyEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality,
minusEquality,
axiomEquality,
imageElimination,
addEquality
Latex:
\mforall{}x,y:\mBbbN{} {}\mrightarrow{} \mBbbR{}. \mforall{}a,b:\mBbbR{}. (lim n\mrightarrow{}\minfty{}.x[n] = a {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.y[n] = b {}\mRightarrow{} lim n\mrightarrow{}\minfty{}.x[n] * y[n] = a * b)
Date html generated:
2017_10_03-AM-09_05_48
Last ObjectModification:
2017_07_28-AM-07_41_44
Theory : reals
Home
Index